I think I'd have replied with something like: "Thank you for taking responsibility and beginning to compensate me for the extra time I worked for your company.

I think I'd have replied with something like: "Thank you for taking responsibility and beginning to compensate me for the extra time I worked for your company. The balance that is owed is $AMOUNT. If you would like to contest this amount, please provide a detailed accounting of your calculations. Otherwise, I will appreciate knowing your intended payment schedule. Again, thank you for acknowledging the former shortcomings with regards to salary and pay. --Former Employee" I would honestly have steered the correspondence in this direction. Kinda saying "yes, the money you sent is MINE" without saying it directly.